Suppose there are three pastures with grass of identical height, density, and growth rate. The first is ten acres and can feed twenty-one horses for nine weeks. The second is three and a third acres and can feed twelve horses for four weeks. The third is twenty-four acres. How many horses can be fed on this third field for eighteen weeks?

This is actually somewhat complex. You need to take into account the fact that, while the grass is being eaten, it is also growing. At some point, the fields are grazed to the point of uselessness, but it isn't just a matter of the grass being eaten; you also have to account for regrowth.

From the first field, we get that 21 horses take 9 weeks to graze the 10 acres down to the ground. During that time, the grass is growing at some rate G cubic centimeters per week per acre. (It's "cubic" because we're taking the area of an acre and multiplying by Since the fields are assumed to start from the same conditions, this means that we have 10 acres that start at some volume, based on the original height, of H cc/acre of grass. The horses eat this at some rate E cc per horse per week. So, after nine weeks at this rate, we have

I think "24H + 432G - 18x(25/27)G = 0" goes to "24(28/3)G+432G-2x(25/3)G=0" and then "224G+432G-(50/3)xG=0". Then x=(656G)/(50G/3)=39.36. But that's still not a whole number. Maybe i should round?

i also tried it this way: the 1st field has 10H+90G of grass that the horses eat in 9 wks, so they each eat (10H+90G)/(189) /wk. the 2nd field has them eating (10H/3+40G/3)/48 /wk. these are the same, so H=12G. then the 3rd field gives (24H+432G)/18x /wk. this gives x=36.

i can't find the mistke doing it the other way but this way works. thanks for the help.